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326 8926. 00097 "bR. in degrees. u. 393, 90). 'AE = E(2'A') - E( 1' A ' ) in kcal/mol. 777 54 BYRON H. LENGSFIELD I11 ANI) DAVID R . YARKONY has been done by Levy and coworkers’ ” using polarized atomic orbitals, or (3) are based on block diagonalization t e c h n i q ~ e s . ’ ~ ~ ” ~ ’ The recently introduced gradient driven techniques for locating actual6’ and/or avoided48 crossings discussed in this chapter should have a significant impact on the field. By enabling efficient determination of regions of significant electronic nonadiabaticity for systems in which molecular point group system is not a determining factor, these techniques have the potential to extend considerably the range of tractable systems.

0, As noted previously the determination of Aa! from the second derivative CP-SAMCSCF equations parallels the deternhation of AFj from the first derivative CP-SAMCSCF equations. The details of the treatment of the second-order CP-SAMCSCF equations can be found in Appendix E and Ref. 38. Note, however, that as in the evaluation of f”(R) the Z-vector method can be used to avoid solving the second order CP-SAMCSCF equations for A;;. 3. A - d y t i c Evaluation of h$(R) Given the formal expression for d;;, it is possible to complete the description of the analytic evaluation of h;i(R) using only previously introduced concepts.

127) into the requirement Eq. 128) which is solved by partial integration. 129) a2 which from Eqs. 8) becomes k$(R) = kii(R). 130) Thus, the methodology used to determine the k"(R) introduced in Section I1 C can be used to study computationally the existence of a rigorous diabatic basis. Note that if in the two-state problem the CSF basis consists of two terms, then Eq. 130) is a consequence of the antisymmetry of f"(R). 93 Finally, note that for diatomic systems for which there is only one internal degree of freedom Eq.